Question

The points \((2,1)\) and \((-3,-4)\) are opposite vertices of a parallelogram. If the other two vertices lie on the line \(x+9y+c=0\), then \(c\) is

• A. 12
• B. 14
• C. 13
• D. 15

Answer 1

The Correct Option is B

The given line also passes through the point of intersection of the diagonals of the parallelogram, which is the mid-point of \((2,1)\) and \((-3,-4)\)The mid-point of the given two points is \(\bigg(-\frac{1}{2},-\frac{3}{2}\bigg)\).
Substituting the point in the given equation \(-\frac{1}{2}+9×-\frac{3}{2}+c=0⇒c=14\)

So, the correct answer is (B): \(14\)

Answer 2

We know that, in a parallelogram the midpoints of two diagonals are the same.
So, the midpoint of \((2,1)\) and \((-3,-4)\) lie on \(x + 9y + c = 0\)
Now, the midpoint of \((2,1)\) and \((-3,-4)\)
\(=(\frac {2-3}{2}, \frac {1-4}{2})\)

\(=(-\frac 12 , -\frac 32)\)

Put \(x= -\frac 12\) and \(y=-\frac 32\) in above equartion,

\(-\frac 12+9(-\frac 32)+c=0\)

\(⇒-\frac 12-\frac {27}{2}+c=0\)

\(⇒c=\frac {28}{2}\)
\(⇒c=14\)

So, the correct option is (B): \(14\)